What is the integral of 2/3 e^xsin(x) ?

The integral of 2/3 e^xsin(x) is 1/3 (-e^xcos(x)+e^xsin(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 2/3 e^xsin(x)

\int \frac{2}{3} e^{x} sin (x) d x

\frac{1}{3} (- e^{x} cos (x) + e^{x} sin (x)) + C

Solution steps

\int \frac{2}{3} e^{x} sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{3} \cdot \int e^{x} sin (x) d x

Apply Integration By Parts

: - e^{x} cos (x) - \int - e^{x} cos (x) d x

= \frac{2}{3} (- e^{x} cos (x) - \int - e^{x} cos (x) d x)

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{3} (- e^{x} cos (x) - (- \int e^{x} cos (x) d x))

Apply Integration By Parts

: e^{x} sin (x) - \int e^{x} sin (x) d x

= \frac{2}{3} (- e^{x} cos (x) - (- (e^{x} sin (x) - \int e^{x} sin (x) d x)))

Therefore

\frac{2}{3} \cdot \int e^{x} sin (x) d x = \frac{2}{3} (- e^{x} cos (x) - (- (e^{x} sin (x) - \int e^{x} sin (x) d x)))

Isolate

\int e^{x} sin (x) d x

= \frac{2}{3} (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2})

Simplify

\frac{2}{3} (- \frac{e ^{x} cos ( x )}{2} + \frac{e ^{x} sin ( x )}{2}) : \frac{1}{3} (- e^{x} cos (x) + e^{x} sin (x))

= \frac{1}{3} (- e^{x} cos (x) + e^{x} sin (x))

Add a constant to the solution

= \frac{1}{3} (- e^{x} cos (x) + e^{x} sin (x)) + C

\int \frac{6 x ^{2}}{4 x ^{6} + 4} d x

\int + cos (3 t) sin (3 t) d t

\int \frac{2 x ^{2} + x - 3}{x ^{3} + 2 x ^{2} - 7 x + 4} d x

\int x^{2} (x - 3)^{10} d x

\int \frac{x ^{2}}{( x ^{3} - 2 ) ^{2}} d x

What is the integral of 2/3 e^xsin(x) ?
The integral of 2/3 e^xsin(x) is 1/3 (-e^xcos(x)+e^xsin(x))+C